| import OAI.Computability.FourierTransform.ChunkCompiler |
|
|
| namespace OAI |
|
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| namespace PowerSaving.Slant |
| open Reel Stroke Zone Spectral Finset |
| noncomputable section |
|
|
| def far (b j : ℕ) := if j<8 then 8*b else 0 |
|
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| def trestle (b j : ℕ) : Zone := ⟨far b j, b⟩ |
| def barrier (b j : ℕ) : Zone := ⟨far b j + b, 2*b⟩ |
| def lockBay (b j : ℕ) : Zone := ⟨far b j + 3*b,2*b⟩ |
|
|
| structure Wharf (p : ℕ) (u v z l m : Zone) : Prop where |
| fitsU : u.Fits p |
| fitsV : v.Fits p |
| fitsZ : z.Fits p |
| fitsL : l.Fits p |
| fitsM : m.Fits p |
| uz : u.Sep z |
| ul : u.Sep l |
| um : u.Sep m |
| lm : l.Sep m |
| zl : z.Sep l |
| zm : z.Sep m |
| uv : u.Sep v |
| zv : z.Sep v |
| vm : v.Sep m |
|
|
| lemma wharf (b j w s : ℕ) (hj : j<16) : |
| Wharf (16*b+w) (quay b j) (jetty (16*b) w b s) |
| (trestle b j) (barrier b j) (lockBay b j) := by |
| let o := far b j |
| have h := quay_fits 16 b j w hj |
| have ho : o+8*b ≤ 16*b := by unfold o far; split <;> omega |
| have bed (v t : ℕ) (hv : o ≤ v) (ht : v+t ≤ o+8*b) : |
| (quay b j).Sep ⟨v,t⟩ := by |
| rcases lt_or_ge j 8 with g|g |
| · apply sep_of |
| change j*b+b ≤ v |
| have H := Nat.mul_le_mul_right b g |
| change (j+1)*b ≤ _ at H; rw [add_mul,one_mul] at H |
| simp only [o,far,ite_eq_left g] at hv |
| omega |
| · apply Zone.Sep.symm; apply sep_of |
| change v+t ≤ j*b |
| have H := Nat.mul_le_mul_right b g |
| simp only [o,far,ite_eq_right (by omega : ¬j<8)] at ht; omega |
| change Wharf (16*b+w) _ _ ⟨o,b⟩ ⟨o+b,2*b⟩ ⟨o+3*b,2*b⟩ |
| refine ⟨h,jetty_fits (16*b) w b s, |
| fits_of _ _ (by change o+b ≤ _; omega),fits_of _ _ (by change o+b+2*b ≤ _; omega), |
| fits_of _ _ (by change o+3*b+2*b ≤ _; omega),bed _ _ le_rfl (by omega),bed _ _ (by omega) |
| (by omega),bed _ _ (by omega) (by omega), |
| sep_of _ _ (by change o+b+2*b ≤ o+3*b; omega), sep_of _ _ le_rfl, |
| sep_of _ _ (by change o+b ≤ o+3*b; omega),quay_sep 16 w b s j hj, ?_, ?_⟩ |
| · apply sep_of |
| change o+b ≤ 16*b+_ |
| omega |
| · apply Zone.Sep.symm; apply sep_of |
| change o+3*b+2*b ≤ 16*b+_ |
| omega |
|
|
| def cliff (k : ℕ) (r : ℂ) (u v z l m : Zone) (b e y : ℕ) (h g : Line) : Reel := |
| cameo (bore k r u z l m b e g) (bore k r z v l m e y h) |
| (bore k r u z l m b e (fun i => -g i)) (bore k r z v l m e y (fun i => -h i)) |
|
|
| structure Slips (k : ℕ) (u v z l m : Zone) : Prop where |
| uw : u.wide ≤ 2^k |
| vw : v.wide ≤ 2^k |
| zw : z.wide ≤ 2^k |
| lw : l.wide=2^(k+1) |
| mw : m.wide=2^(k+1) |
|
|
| lemma slips (k j w s : ℕ) : |
| let b := 2^k |
| Slips k (quay b j) (jetty (16*b) w b s) (trestle b j) |
| (barrier b j) (lockBay b j) := by |
| intro b |
| refine ⟨le_rfl,min_le_left _ _,le_rfl,(pow_succ' _ _).symm,(pow_succ' _ _).symm⟩ |
|
|
| section |
| variable {u v z l m : Zone} {k n : ℕ} (r : ℂ) (b e y : ℕ) (h g : Line) |
| (H : Slips k u v z l m) (hf : Wharf n u v z l m) |
|
|
| include H hf |
|
|
| lemma sound_cliff : |
| (cliff k r u v z l m b e y h g).Sound n := by |
| have ha (g : Line) := sound_bore r b e g H.uw H.zw H.lw H.mw hf.ul hf.lm hf.zm |
| hf.fitsU hf.fitsZ hf.fitsL hf.fitsM |
| have hb (g : Line) := sound_bore r e y g H.zw H.vw H.lw H.mw hf.zl hf.lm hf.vm |
| hf.fitsZ hf.fitsV hf.fitsL hf.fitsM |
| exact sound_cameo _ _ _ _ n (ha _) (hb _) (ha _) (hb _) |
|
|
| lemma spill_cliff (hr : IsPrimitiveRoot r (2^(k+1))) : |
| Spill (cliff k r u v z l m b e y h g) u v |
| (spout z.wide e y h ∘ spout u.wide b e g) := by |
| have ha (g : Line) := spill_bore r b e g H.uw H.zw H.lw H.mw hf.ul hf.lm hf.zm |
| hr hf.um hf.uz |
| have hb (h : Line) := spill_bore r e y h H.zw H.vw H.lw H.mw hf.zl hf.lm hf.vm |
| hr hf.zm hf.zv |
| apply spill_cameo (ha g) (hb h) |
| · rw [← spout_neg]; exact ha _ |
| · rw [← spout_neg]; exact hb _ |
| · exact reads_spout _ b e _ g |
| · exact reads_spout _ e y _ h |
| · exact additive_spout _ e y _ h |
| · exact hf.uv |
| · exact hf.uz |
| · exact hf.zv |
|
|
| end |
|
|
| def gravel (k : ℕ) (r : ℂ) (w : ℕ) (h g : Line) (s c j : ℕ) : Reel := |
| let b := 2^k |
| let p := 16*b |
| cliff k r (quay b j) (jetty p w b s) (trestle b j) (barrier b j) |
| (lockBay b j) (j*b) (c*b) (p+s*b) h g |
|
|
| def reef (k : ℕ) (r : ℂ) (w : ℕ) (h g : Line) : Reel := |
| riprap (gravel k r w h g) 16 |
|
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| lemma beats_gravel (k : ℕ) (r : ℂ) (w : ℕ) (h g : Line) (s c j : ℕ) : |
| (gravel k r w h g s c j).beats=4*(80*k+348) := by |
| unfold gravel cliff |
| dsimp only |
| simp only [beats_cameo,beats_bore]; omega |
| lemma beats_reef (k : ℕ) (r : ℂ) (w : ℕ) (h g : Line) : |
| (reef k r w h g).beats=16*(16*(16*(4*(80*k+348)))) := |
| beats_riprap _ _ _ (fun _ _ _ _ _ _ => beats_gravel k _ _ _ _ _ _ _) |
|
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| lemma sound_gravel (k : ℕ) (r : ℂ) (w : ℕ) (h g : Line) (s c j : ℕ) (hj : j<16) : |
| (gravel k r w h g s c j).Sound (16*2^k+w) := |
| sound_cliff r (j*_ ) (c*_) (16*2^k+s*_) h g (slips k j w s) (wharf (2^k) j w s hj) |
| lemma sound_reef (k : ℕ) (r : ℂ) (w : ℕ) (h g : Line) : |
| (reef k r w h g).Sound (16*2^k+w) := |
| sound_riprap _ _ _ (fun _ _ _ _ j hj => sound_gravel k _ _ _ _ _ _ j hj) |
|
|
| lemma spill_gravel (k : ℕ) (r : ℂ) (w : ℕ) (h g : Line) (s c j : ℕ) (hj : j<16) |
| (hr : IsPrimitiveRoot r (2^(k+1))) : |
| Spill (gravel k r w h g s c j) (quay (2^k) j) |
| (jetty (16*2^k) w (2^k) s) (wave (16*2^k) (2^k) s c j h g) := |
| spill_cliff r (j*2^k) (c*2^k) (16*2^k+s*2^k) h g (slips k j w s) |
| (wharf (2^k) j w s hj) hr |
| lemma spill_reef (k : ℕ) (r : ℂ) (w : ℕ) (h g : Line) (hw : w ≤ 16*2^k) |
| (hr : IsPrimitiveRoot r (2^(k+1))) : |
| Spill (reef k r w h g) (initial (16*2^k)) (coast (16*2^k) w) (surf (16*2^k) h g) := |
| spill_riprap _ 16 _ w h g (by positivity) hw |
| (fun s _ c _ j hj => spill_gravel k r w h g s c j hj hr) |
|
|
| -- b=1 small cross |
| def speck (p w : ℕ) (h g : Line) (s c j : ℕ) : Reel := |
| if s<w then flash (jetty p w 1 s) (quay 1 j) (fun _ => rise h (p+s) c*rise g c j) |
| else wait 26 |
|
|
| def sand (p w : ℕ) (h g : Line) : Reel := riprap (speck p w h g) p |
|
|
| lemma beats_speck (p w : ℕ) (h g : Line) (s c j : ℕ) : |
| (speck p w h g s c j).beats=26 := by unfold speck; split; exact beats_flash _ _ _; rfl |
| lemma beats_sand (p w : ℕ) (h g : Line) : (sand p w h g).beats=p*(p*(p*26)) := |
| beats_riprap _ _ _ fun _ _ _ _ _ _ => beats_speck _ _ _ _ _ _ _ |
|
|
| lemma quay_jet (p w j s : ℕ) (hj : j<p) : |
| (jetty p w 1 s).Sep (quay 1 j) := by |
| simpa only [mul_one] using (quay_sep p w 1 s j hj).symm |
| lemma sound_speck (p w : ℕ) (h g : Line) (s c j : ℕ) (hj : j<p) : |
| (speck p w h g s c j).Sound (p+w) := by |
| unfold speck; split |
| · rename_i hs |
| refine sound_flash (jetty_fits p w 1 s) ?_ ?_ (quay_jet p w j s hj) _ |
| · simpa only [mul_one] using quay_fits p 1 j w hj |
| · change min 1 (w-s*1) = 1; omega |
| · exact sound_wait 26 _ |
| lemma sound_sand (p w : ℕ) (h g : Line) : (sand p w h g).Sound (p+w) := |
| sound_riprap _ _ _ fun _ _ _ _ j hj => sound_speck _ _ _ _ _ _ j hj |
|
|
| lemma spill_speck (p w : ℕ) (h g : Line) (s c j : ℕ) (hj : j<p) : |
| Spill (speck p w h g s c j) (quay 1 j) (jetty p w 1 s) (wave p 1 s c j h g) := by |
| unfold speck |
| split |
| · rename_i hs |
| have hh : (jetty p w 1 s).wide = (quay 1 j).wide := by |
| change min 1 (w-s*1)=1; omega |
| refine (spill_flash hh (quay_jet p w j s hj) _).congr ?_ |
| intro x i hi |
| rw [hh] at hi |
| change i<1 at hi |
| have h0 : i=0 := by omega |
| simp only [wave,Function.comp_apply,h0,spout,sum_range_one,mul_one,add_zero,mul_assoc] |
| · rename_i hs |
| constructor |
| · intro x i hi |
| change i < min 1 (w-s*1) at hi; omega |
| · intro x i hi; rw [go_wait] |
| lemma spill_sand (p w : ℕ) (h g : Line) (hw : w ≤ p) : |
| Spill (sand p w h g) (initial p) (coast p w) (surf p h g) := by |
| have hh s (_ : s<p) c (_ : c<p) j (hj : j<p) : |
| Spill (speck p w h g s c j) (quay 1 j) (jetty (p*1) w 1 s) (wave (p*1) 1 s c j h g) := |
| by simpa only [mul_one] using spill_speck p w h g s c j hj |
| unfold sand |
| simpa only [mul_one] using |
| spill_riprap (speck p w h g) p 1 w h g (by omega) (by omega) hh |
|
|
| end |
| end PowerSaving.Slant |
|
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| end OAI |
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